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Constructions and loci

Use ruler and compass constructions accurately, then interpret loci as sets of points that satisfy distance rules.

Geometry and MeasuresFoundation and HigherGrades 4 to 7Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Constructions and loci

Pearson Edexcel GCSE Maths geometry G2: use standard ruler and compass constructions and construct loci.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A construction is an accurate drawing made using a ruler, compass and pencil. Do not estimate by eye unless the question says to measure.

A perpendicular bisector cuts a line segment in half at 90 degrees. It is the locus of points that are the same distance from both endpoints.

To construct a perpendicular bisector, keep the same compass width from both endpoints and draw arcs above and below the line. Join the arc intersections.

An angle bisector splits an angle into two equal angles. It is the locus of points that are the same distance from the two arms of the angle.

A locus is a path or region of points that follow a rule, such as all points 3 cm from A or all points less than 2 cm from a line.

For points a fixed distance from a point, draw a circle. For points a fixed distance from a straight line, draw two parallel lines.

For points closer to A than B, use the perpendicular bisector of AB as the boundary and shade the side containing A.

In exam questions, leave construction arcs visible unless told otherwise. They are often evidence for method marks.

Key ruleConstructions must show compass arcs. Loci describe all points that satisfy a distance rule.

Diagram guide

AB90 degreesequal compass arcs meet above and belowperpendicular bisector of AB
Perpendicular bisectorEqual arcs from both endpoints meet above and below the line. Joining them gives a 90-degree bisector.
vertexbisectorsame compass width from both armsangle is split into two equal parts
Angle bisectorUse equal compass arcs from the two arms of the angle, then draw a line from the vertex through the arc intersection.
all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
Loci regionsA locus can be a line, circle, boundary or shaded region depending on the distance rule.

Worked examples

Construct a perpendicular bisector

Construct the perpendicular bisector of line segment AB.

AB90 degreesequal compass arcs meet above and belowperpendicular bisector of AB
Example: perpendicular bisectorKeep the compass width greater than half of AB.
  1. Open the compass to more than half the length of AB.
  2. From A, draw arcs above and below the line.
  3. Without changing the compass width, repeat from B.
  4. Join the two arc intersections with a straight line.

Answer: A line crossing AB at 90 degrees and splitting AB into two equal parts.

Construct an angle bisector

Construct the bisector of an angle.

vertexbisectorsame compass width from both armsangle is split into two equal parts
Example: angle bisectorThe bisector divides the angle into two equal angles.
  1. Place the compass on the vertex and draw an arc crossing both arms.
  2. From each crossing point, draw arcs inside the angle using the same compass width.
  3. Join the vertex to the intersection of the two new arcs.

Answer: A ray from the vertex that splits the angle into two equal parts.

Locus from a point

Draw the locus of points exactly 4 cm from point A.

all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
Example: fixed distance from a pointAll points the same distance from one point form a circle.
  1. Set the compass to 4 cm.
  2. Place the compass point on A.
  3. Draw a full circle around A.

Answer: A circle centre A with radius 4 cm.

Locus from a line

Shade the region less than 2 cm from line l.

all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
Example: within a distance of a linePoints within 2 cm of a line lie between two parallel boundary lines.
  1. Draw one line parallel to l, 2 cm on one side.
  2. Draw a second parallel line 2 cm on the other side.
  3. Shade the strip between the two boundary lines.

Answer: The shaded strip within 2 cm of the line.

Common mistakes

  • Rubbing out compass arcs, which can remove evidence for construction marks.
  • Using a protractor for a construction when compass arcs are required.
  • Drawing only one arc for a perpendicular bisector.
  • Confusing a fixed distance from a point with a fixed distance from a line.
  • Shading the wrong side of a locus boundary.

Quick exercise

Try these before moving to the exam-style questions.

  1. What construction cuts a line segment in half at 90 degrees?
    AB90 degreesequal compass arcs meet above and belowperpendicular bisector of AB
    Quick check: line segmentThe line is both perpendicular and a bisector.
  2. What shape is the locus of points 5 cm from point A?
    all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
    Quick check: fixed distance from a pointA fixed distance from one point creates a circle.
  3. What construction splits an angle into two equal angles?
    vertexbisectorsame compass width from both armsangle is split into two equal parts
    Quick check: equal anglesThe angle bisector is the locus of points equally far from the two arms.
  4. What do you draw for points exactly 3 cm from a straight line?
    all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
    Quick check: fixed distance from a lineDraw parallel boundary lines at the fixed distance.
  5. For points closer to A than B, which construction gives the boundary?
    AB90 degreesequal compass arcs meet above and belowperpendicular bisector of AB
    Quick check: closer to A than BThe perpendicular bisector separates points nearer A from points nearer B.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationNon-calculator3 marks

Construct the perpendicular bisector of line segment AB.

AB90 degreesequal compass arcs meet above and belowperpendicular bisector of AB
Question diagram: perpendicular bisectorShow equal compass arcs from A and B.
constructionsperpendicular bisectorruler and compass
Standard exam styleFoundation and HigherNon-calculator3 marks

A point P must be exactly 4 cm from A and exactly 4 cm from B. Describe how to find possible positions for P.

all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
Question diagram: intersecting lociDraw two circles and use their intersection points.
lociintersectionsdistance from a point
ChallengeHigherNon-calculator4 marks

Shade the region of points that are less than 3 cm from line l and closer to A than B.

all points 3 cmfrom point AAline lwithin 2 cm of a lineshade the region that satisfies the conditions
Question diagram: combined lociUse the distance-from-line boundary and the perpendicular bisector boundary.
combined locilocus regionhigher geometry